Weak Convergence of Random Functions Defined by The Eigenvectors of Sample Covariance Matrices

Abstract

Let $\{v_{ij}\}, i, j = 1, 2, \ldots,$ be i.i.d. symmetric random variables with $\mathbb{E}(\nu^4_{11}) < \infty$, and for each $n$ let $M_n = (1/s)V_n V^T_n$, where $V_n = (v_{ij}), i = 1, 2, \ldots, n, j = 1, 2, \ldots, s = s(n)$ and $n/s \rightarrow y > 0$ as $n \rightarrow \infty$. Denote by $O_n \Lambda_n O^T_n$ the spectral decomposition of $M_n$. Define $X \in D\lbrack 0, 1 \rbrack$ by $X_n(t) = \sqrt{n/2} \sum^{\lbrack nt \rbrack}_{i = 1}(y^2_i - 1/n)$ where $(y_1, y_2, \ldots, y_n)^T = O^T(\pm 1/\sqrt{n}, \pm 1/ \sqrt{n}, \ldots, \pm 1/\sqrt{n})^T$. It is shown that $X_n \rightarrow_\mathscr{D} W^0$ as $n \rightarrow \infty$, where $W^0$ is a Brownian bridge. This result sheds some light on the problem of describing the behavior of the eigenvectors of $M_n$ for $n$ large and for general $v_{11}$.